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Theorem onsucuni2 6668
Description: A successor ordinal is the successor of its union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
onsucuni2  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  suc  U. A  =  A )

Proof of Theorem onsucuni2
StepHypRef Expression
1 eleq1 2529 . . . . . 6  |-  ( A  =  suc  B  -> 
( A  e.  On  <->  suc 
B  e.  On ) )
21biimpac 486 . . . . 5  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  suc  B  e.  On )
3 eloni 4897 . . . . 5  |-  ( suc 
B  e.  On  ->  Ord 
suc  B )
4 ordsuc 6648 . . . . . . . 8  |-  ( Ord 
B  <->  Ord  suc  B )
5 ordunisuc 6666 . . . . . . . 8  |-  ( Ord 
B  ->  U. suc  B  =  B )
64, 5sylbir 213 . . . . . . 7  |-  ( Ord 
suc  B  ->  U. suc  B  =  B )
7 suceq 4952 . . . . . . 7  |-  ( U. suc  B  =  B  ->  suc  U. suc  B  =  suc  B )
86, 7syl 16 . . . . . 6  |-  ( Ord 
suc  B  ->  suc  U. suc  B  =  suc  B
)
9 ordunisuc 6666 . . . . . 6  |-  ( Ord 
suc  B  ->  U. suc  suc 
B  =  suc  B
)
108, 9eqtr4d 2501 . . . . 5  |-  ( Ord 
suc  B  ->  suc  U. suc  B  =  U. suc  suc 
B )
112, 3, 103syl 20 . . . 4  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  suc  U. suc  B  =  U. suc  suc  B
)
12 unieq 4259 . . . . . 6  |-  ( A  =  suc  B  ->  U. A  =  U. suc  B )
13 suceq 4952 . . . . . 6  |-  ( U. A  =  U. suc  B  ->  suc  U. A  =  suc  U. suc  B
)
1412, 13syl 16 . . . . 5  |-  ( A  =  suc  B  ->  suc  U. A  =  suc  U.
suc  B )
15 suceq 4952 . . . . . 6  |-  ( A  =  suc  B  ->  suc  A  =  suc  suc  B )
1615unieqd 4261 . . . . 5  |-  ( A  =  suc  B  ->  U. suc  A  =  U. suc  suc  B )
1714, 16eqeq12d 2479 . . . 4  |-  ( A  =  suc  B  -> 
( suc  U. A  = 
U. suc  A  <->  suc  U. suc  B  =  U. suc  suc  B ) )
1811, 17syl5ibr 221 . . 3  |-  ( A  =  suc  B  -> 
( ( A  e.  On  /\  A  =  suc  B )  ->  suc  U. A  =  U. suc  A ) )
1918anabsi7 819 . 2  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  suc  U. A  = 
U. suc  A )
20 eloni 4897 . . . 4  |-  ( A  e.  On  ->  Ord  A )
21 ordunisuc 6666 . . . 4  |-  ( Ord 
A  ->  U. suc  A  =  A )
2220, 21syl 16 . . 3  |-  ( A  e.  On  ->  U. suc  A  =  A )
2322adantr 465 . 2  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  U. suc  A  =  A )
2419, 23eqtrd 2498 1  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  suc  U. A  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   U.cuni 4251   Ord word 4886   Oncon0 4887   suc csuc 4889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-tr 4551  df-eprel 4800  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-suc 4893
This theorem is referenced by:  rankxplim3  8316  rankxpsuc  8317
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